Can Optimal Rearrangement Invariant Sobolev Imbeddings be Further Extended?

ABSTRACT. Sobolev imbeddings (over suitable open subsets of Rn) can be extended from the classical Lp-setting to that of more gen-eral norms (required to be rearrangement invariant) on the underlying function spaces. This has been thoroughly studied in recent years and shown to be intimately connected to an associated kernel operator (of one variable). This kernel operator always has an optimal domain (being a Banach function space, but typically not rearrangement in-variant) to which it can be continuously extended. So, techniques aside, there is no a priori reason not to treat Sobolev imbeddings for non-rearrangement invariant norms. This is the aim of the present paper. 1